Subgroups — #25013662

Order 11008.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063270	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582993	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582702	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582699	C₂	1	1	yes	maximal, characteristic
5	#19135459	C₂	1	1	yes	maximal, characteristic
6	#19132672	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063290	—	2	1	no	—
11	C₂×#13063270	—	2	2	no	—
12	C₂×#13063270	—	2	2	no	—
13	C₂×#13063270	C₄	1	4	yes	complemented
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	#14582993	—	2	2	no	—
16	#14582993	—	2	2	no	—
17	C₂×#13063248	—	2	1	no	—
18	C₂×#13063245	—	2	1	no	—
19	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	#14582800	—	2	1	no	—
21	#14582702	—	2	2	no	—
22	#14582699	—	2	2	no	—
23	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₈₆	—	2	1	no	—
27	C₂²×D₁₇₂	—	2	1	no	—
28	C₂×Dic₁₇₂	—	2	1	no	—
29	C₂×Dic₁₇₂	—	2	1	no	—
30	C₂×Dic₁₇₂	—	2	1	no	—
31	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
32	C₂×Dic₁₇₂	D₄	1	2	yes	—
33	C₂×Dic₁₇₂	D₄	1	2	yes	—
34	C₂×D₃₄₄	—	2	1	no	—
35	C₂×D₃₄₄	—	2	1	no	—
36	C₂×D₃₄₄	—	2	1	no	—
37	C₂×D₃₄₄	—	2	1	no	—
38	C₂×D₃₄₄	—	2	1	no	—
39	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
40	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
41	#13063290	—	2	4	no	—
42	#13063290	—	4	2	no	—
43	C₂×Dic₄₃⋊C₄	—	4	1	no	—
44	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
45	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
46	C₂×C₄×Dic₄₃	—	4	1	no	—
47	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
48	#13063270	—	2	8	no	—
49	C₂×C₄₃⋊C₁₆	—	2	1	no	—
50	C₂×C₄₃⋊C₁₆	—	2	1	no	—
51	C₂×C₄₃⋊C₁₆	—	2	1	no	—
52	#13063248	—	2	4	no	—
53	#13063245	—	4	2	no	—
54	C₈×Dic₄₃	—	2	1	no	—
55	C₈×Dic₄₃	—	2	1	no	—
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₁₆×C₄₃	—	2	1	no	—
60	C₂×C₁₆×C₄₃	—	2	1	no	—
61	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
62	#107×C₄₃	—	2	1	no	—
63	#107×C₄₃	—	2	1	no	—
64	#107×C₄₃	—	2	1	no	—
65	#107×C₄₃	D₄	1	2	yes	—
66	C₂³×D₄₃	—	4	1	no	—
67	C₂²×C₄×C₄₃	—	4	1	no	—
68	C₂²×C₄×C₄₃	—	4	1	no	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	4	1	no	—
71	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
72	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
73	C₂×D₁₇₂	—	2	2	no	—
74	C₂×D₁₇₂	—	4	1	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×D₁₇₂	—	4	1	no	—
77	C₂×D₁₇₂	—	4	1	no	—
78	C₂×D₁₇₂	—	4	1	no	—
79	C₂×Dic₈₆	—	2	2	no	—
80	C₂×Dic₈₆	—	2	2	no	—
81	C₂×Dic₈₆	—	2	2	no	—
82	C₂×Dic₈₆	—	4	1	no	—
83	C₂×Dic₈₆	—	4	1	no	—
84	C₂×Dic₈₆	—	4	1	no	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
90	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	—	4	1	no	—
93	C₄⋊C₄×C₄₃	—	4	1	no	—
94	C₄⋊C₄×C₄₃	—	4	1	no	—
95	(C₂×D₄₃)⋊C₄	—	8	2	no	—
96	(C₂×D₄₃)⋊C₄	—	8	2	no	—
97	Dic₄₃⋊C₄	—	4	4	no	—
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	4	4	no	—
101	C₂×C₄₃⋊C₈	—	2	1	no	—
102	C₂×C₄₃⋊C₈	—	2	1	no	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	Dic₁₇₂	—	2	2	no	—
105	Dic₁₇₂	—	2	2	no	—
106	Dic₁₇₂	—	2	2	no	—
107	Dic₁₇₂	C₂²⋊C₄	1	4	yes	complemented
108	D₃₄₄	—	2	2	no	—
109	D₃₄₄	—	2	2	no	—
110	D₃₄₄	—	4	1	no	—
111	D₃₄₄	—	4	1	no	—
112	C₁₆×C₄₃	—	2	2	no	—
113	C₄₃⋊C₁₆	—	4	1	no	—
114	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₄₃	—	4	2	no	—
116	C₂²×D₄₃	—	8	1	no	—
117	C₂²×D₄₃	—	8	1	no	—
118	C₂²×D₄₃	—	8	1	no	—
119	C₂²×D₄₃	—	8	1	no	—
120	C₂²×D₄₃	—	8	1	no	—
121	C₂²×D₄₃	—	8	1	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	8	1	no	—
124	C₂×C₄×C₄₃	—	8	1	no	—
125	C₂×C₄×C₄₃	—	8	1	no	—
126	C₂×C₄×C₄₃	—	8	1	no	—
127	C₂×C₄×C₄₃	—	8	1	no	—
128	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
132	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×Dic₄₃	—	2	1	no	—
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	2	1	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	SD₃₂	1	2	yes	complemented
143	D₁₇₂	—	4	2	no	—
144	D₁₇₂	—	4	2	no	—
145	D₁₇₂	—	8	1	no	—
146	D₁₇₂	—	8	1	no	—
147	Dic₈₆	—	2	4	no	—
148	Dic₈₆	—	4	2	no	—
149	Dic₈₆	—	4	2	no	—
150	Dic₈₆	—	4	2	no	—
151	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
153	C₈×C₄₃	#121	1	1	yes	characteristic
154	C₈×C₄₃	#123	1	1	yes	characteristic
155	C₄₃⋊C₈	—	4	1	no	—
156	#13013	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
158	C₂²×C₄₃	#447	1	1	yes	characteristic
159	C₂²×C₄₃	#459	1	1	yes	characteristic
160	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
161	C₂²×C₄₃	#508	1	1	yes	characteristic
162	C₂²×C₄₃	#509	1	1	yes	characteristic
163	C₂²×C₄₃	#510	1	1	yes	characteristic
164	C₂×D₄₃	—	8	2	no	—
165	C₂×D₄₃	—	8	2	no	—
166	C₂×D₄₃	—	8	2	no	—
167	C₂×D₄₃	—	16	1	no	—
168	C₂×D₄₃	—	16	1	no	—
169	C₂×D₄₃	—	16	1	no	—
170	C₂×D₄₃	—	16	1	no	—
171	C₄×C₄₃	—	8	2	no	—
172	C₄×C₄₃	—	16	1	no	—
173	C₄×C₄₃	#386	1	1	yes	characteristic
174	C₄×C₄₃	#442	1	1	yes	characteristic
175	C₄×C₄₃	#449	1	1	yes	characteristic
176	C₄×C₄₃	#463	1	1	yes	characteristic
177	Dic₄₃	—	2	2	no	—
178	Dic₄₃	—	4	4	no	—
179	Dic₄₃	—	4	1	no	—
180	C₂²×SD₃₂	—	43	1	no	—
181	C₂×C₄×Q₁₆	—	43	1	no	—
182	C₂×#358	—	43	1	no	—
183	C₂×#357	—	43	1	no	—
184	#1970	—	43	1	no	—
185	#1380	D₄₃	1	1	yes	characteristic, complemented
186	#1374	—	43	1	no	—
187	C₂×C₄₃	#1879	1	1	yes	characteristic
188	C₂×C₄₃	C₄×SD₃₂	1	1	yes	characteristic
189	C₂×C₄₃	#2175	1	1	yes	characteristic
190	C₂×C₄₃	#2185	1	1	yes	characteristic
191	C₂×C₄₃	#2186	1	1	yes	characteristic
192	C₂×C₄₃	#2207	1	1	yes	characteristic
193	C₂×C₄₃	#2211	1	1	yes	characteristic
194	D₄₃	—	16	2	no	—
195	C₂²×Q₁₆	—	43	1	no	—
196	C₂²×D₈	—	43	1	no	—
197	C₂×C₄×Q₈	—	86	1	no	—
198	C₂×SD₃₂	—	43	4	no	—
199	C₂×SD₃₂	—	86	2	no	—
200	C₂×SD₃₂	—	86	2	no	—
201	C₂²×C₁₆	—	43	1	no	—
202	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
203	C₄×Q₁₆	—	86	2	no	—
204	C₄×Q₁₆	—	86	2	no	—
205	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
206	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
207	C₂×#103	—	86	1	no	—
208	C₂×#102	—	86	1	no	—
209	C₂×C₄×C₈	—	43	1	no	—
210	#390	—	86	1	no	—
211	#358	—	86	2	no	—
212	#357	—	86	2	no	—
213	C₄₃	#13013	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	86	1	no	—
215	C₂²×D₄	—	86	1	no	—
216	C₂×Q₁₆	—	43	2	no	—
217	C₂×Q₁₆	—	43	2	no	—
218	C₂×Q₁₆	—	43	2	no	—
219	C₂×Q₁₆	—	86	1	no	—
220	C₂×Q₁₆	—	86	1	no	—
221	C₂×Q₁₆	—	86	1	no	—
222	C₂×D₈	—	43	2	no	—
223	C₂×D₈	—	86	1	no	—
224	C₂×D₈	—	86	1	no	—
225	C₂×D₈	—	86	1	no	—
226	C₂×D₈	—	86	1	no	—
227	C₂×D₈	—	86	1	no	—
228	C₂²×C₈	—	43	1	no	—
229	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
230	C₄×Q₈	—	86	4	no	—
231	C₄×Q₈	—	172	2	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	172	1	no	—
235	C₂×C₂²⋊C₄	—	172	1	no	—
236	C₂×C₂²⋊C₄	—	172	1	no	—
237	C₂×C₄²	—	43	1	no	—
238	C₂×C₄²	—	172	1	no	—
239	SD₃₂	—	86	8	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	2	1	no	—
242	C₂×C₁₆	—	86	1	no	—
243	C₂×C₁₆	—	86	1	no	—
244	C₂×C₁₆	—	86	1	no	—
245	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
246	#107	—	2	1	no	—
247	#107	—	2	1	no	—
248	#107	—	2	1	no	—
249	#107	D₁₇₂	1	2	yes	—
250	#103	—	86	4	no	—
251	#102	—	172	2	no	—
252	C₄×C₈	—	86	1	no	—
253	C₄×C₈	—	86	1	no	—
254	C₂⁴	—	172	1	no	—
255	C₂×Q₈	—	86	2	no	—
256	C₂×Q₈	—	86	2	no	—
257	C₂×Q₈	—	86	2	no	—
258	C₂×Q₈	—	172	1	no	—
259	C₂×Q₈	—	172	1	no	—
260	C₂×Q₈	—	172	1	no	—
261	C₂×D₄	—	86	2	no	—
262	C₂×D₄	—	172	1	no	—
263	C₂×D₄	—	172	1	no	—
264	C₂×D₄	—	172	1	no	—
265	C₂×D₄	—	172	1	no	—
266	C₂×D₄	—	172	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	43	1	no	—
270	C₂²×C₄	—	43	1	no	—
271	C₂²×C₄	—	172	1	no	—
272	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
273	Q₁₆	—	43	4	no	—
274	Q₁₆	—	86	2	no	—
275	Q₁₆	—	86	2	no	—
276	Q₁₆	—	86	2	no	—
277	D₈	—	86	2	no	—
278	D₈	—	86	2	no	—
279	D₈	—	172	1	no	—
280	D₈	—	172	1	no	—
281	C₂×C₈	—	86	1	no	—
282	C₂×C₈	—	86	1	no	—
283	C₂×C₈	—	86	1	no	—
284	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
285	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
286	C₂×C₈	#10621898	1	1	yes	characteristic
287	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
288	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
289	C₂×C₈	#10621902	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	172	4	no	—
295	C₂²⋊C₄	—	344	2	no	—
296	C₂²⋊C₄	—	344	2	no	—
297	C₄²	—	86	1	no	—
298	C₄²	—	86	1	no	—
299	C₄²	—	172	4	no	—
300	C₁₆	—	2	2	no	—
301	C₁₆	—	172	1	no	—
302	C₂³	—	344	1	no	—
303	C₂³	—	344	1	no	—
304	C₂³	—	344	1	no	—
305	C₂³	—	344	1	no	—
306	C₂³	—	344	1	no	—
307	C₂³	—	344	1	no	—
308	C₂³	—	172	2	no	—
309	C₂³	D₈×D₄₃	1	1	yes	characteristic
310	Q₈	—	86	4	no	—
311	Q₈	—	172	2	no	—
312	Q₈	—	172	2	no	—
313	Q₈	—	172	2	no	—
314	D₄	—	344	1	no	—
315	D₄	—	344	1	no	—
316	D₄	—	172	2	no	—
317	D₄	—	172	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	—	8	1	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	8	1	no	—
323	C₂×C₄	—	8	1	no	—
324	C₂×C₄	—	43	2	no	—
325	C₂×C₄	—	86	1	no	—
326	C₂×C₄	—	86	1	no	—
327	C₂×C₄	—	86	1	no	—
328	C₂×C₄	—	86	1	no	—
329	C₂×C₄	—	86	1	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	—	172	2	no	—
332	C₂×C₄	—	172	2	no	—
333	C₂×C₄	#13063287	1	1	yes	characteristic
334	C₂×C₄	#13063295	1	1	yes	characteristic
335	C₂×C₄	#13063298	1	1	yes	characteristic
336	C₂×C₄	#13063348	1	1	yes	characteristic
337	C₂×C₄	#13063352	1	1	yes	characteristic
338	C₂×C₄	#13063352	1	1	yes	characteristic
339	C₈	—	172	1	no	—
340	C₈	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
341	C₈	C₄×D₁₇₂	1	1	yes	characteristic
342	C₈	#13063300	1	1	yes	characteristic
343	C₈	#13063310	1	1	yes	characteristic
344	C₂²	—	344	2	no	—
345	C₂²	—	344	2	no	—
346	C₂²	—	344	2	no	—
347	C₂²	—	688	1	no	—
348	C₂²	—	688	1	no	—
349	C₂²	—	688	1	no	—
350	C₂²	—	688	1	no	—
351	C₂²	#14582981	1	1	yes	characteristic
352	C₂²	#14583034	1	1	yes	characteristic
353	C₂²	#14583093	1	1	yes	characteristic
354	C₂²	SD₃₂×D₄₃	1	1	yes	characteristic
355	C₂²	#14583330	1	1	yes	characteristic
356	C₂²	#14583333	1	1	yes	characteristic
357	C₂²	#14583341	1	1	yes	characteristic
358	C₄	—	8	2	no	—
359	C₄	—	16	1	no	—
360	C₄	—	86	2	no	—
361	C₄	—	172	1	no	—
362	C₄	—	172	4	no	—
363	C₄	#14582777	1	1	yes	characteristic
364	C₄	#14583010	1	1	yes	characteristic
365	C₄	#14583047	1	1	yes	characteristic
366	C₄	#14583117	1	1	yes	characteristic
367	C₂	—	688	2	no	—
368	C₂	#19134944	1	1	yes	characteristic
369	C₂	#19136446	1	1	yes	characteristic
370	C₂	#19136458	1	1	yes	characteristic
371	C₂	#19136513	1	1	yes	characteristic
372	C₂	#19136519	1	1	yes	characteristic
373	C₂	#19136634	1	1	yes	characteristic
374	C₂	#19136660	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1380	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1380	D₄₃
hypocenter	C₄₃	#13013
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#14582993	C₂	semidirect
3. C₂×#14582702	C₂	semidirect
6. #19132672	C₂	semidirect
7. C₄₃×#1380	C₂	semidirect
13. C₂×#13063270	C₄	semidirect
107. Dic₁₇₂	C₂²⋊C₄	semidirect
142. C₂×Dic₄₃	SD₃₂	semidirect
185. #1380	D₄₃	semidirect
213. C₄₃	#13013	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13063270	2
C₂×#14582993	2
C₂×#14582702	2
C₂×#14582699	2
#19135459	2
#19132672	2
C₄₃×#1380	2
#13013	43